Generalized Brillouin condition

In this form, the amplitude equations (21) have been previously studied by Kutzelnigg and named the generalized Brillouin conditions [38]. 

Finally, we require for both the orbital and configurational variation parameters that 0 > satisfies the generalized Brillouin condition... 

These requirements on the matrix elements are precisely the generalized Brillouin conditions for the PHF functions [5]. They are very general and hold also for all the DODS type funtions such as the HPHF function, as well as for the RHF function. 

Eqs. (130)-(132), derived with the assumption of the generalized Brillouin condition [31], entirely define the CCSD(F12) model. It is worth to mention that this model is an approximation to the full CCSD-F12 method [41], where more explicitly correlated terms... 

The MP2-F12 coupling matrix (7 was obtained without the assumption of the generalized Brillouin condition [49]... 

Equation (105) has been referred to as generalized Brillouin condition and is basically the same assumption on which the approximation (33) in Section 2.3 is based. In practice, the extended and generalized Brillouin conditions require that a one-particle basis set of near-Hartree-Fbck limit quality must be used. By virtue of the generalized Brillouin condition, the Lt term reduces to... 

GENERALIZED NORMAL ORDERING, IRREDUCIBLE BRILLOUIN CONDITIONS, AND CONTRACTED SCHRODINGER EQUATIONS... 

Formulating conditions for the energy to be stationary with respect to variations of the wavefunction P in this generalized normal ordering, one is led to the irreducible Brillouin conditions and irreducible contracted Schrodinger equations, which are conditions on the one-particle density matrix and the fe-particle cumulants k, and which differ from their traditional counterparts (even after reconstruction [4]) in being strictly separable (size consistent) and describable in terms of connected diagrams only. 

While the (one-particle) Brillouin condition BCi has been known for a long time, and has played a central role in Hartree-Fock theory and in MC-SCF theory, the generalizations for higher particle rank were only proposed in 1979 [38], although a time-dependent formulation by Thouless [39] from 1961 can be regarded as a precursor. 

If one formulates the conditions for stationarity of the energy expectation value in terms of generalized normal ordering, one is led to either the irreducible fc-particle Brillouin conditions IBCj or the irreducible A -particle contracted Schrodinger equations (IBC ), which are conditions to be satisfied by y = yj and the k. One gets a hierarchy of k-particle approximations that can be truncated at any desired order, without any need for a reconstruction, as is required for the reducible counterparts. 

W. Kutzelnigg, Generalized k-particle Brillouin conditions and their use for the construction of correlated electronic wave functions. Chem. Phys. Lett. 64, 383 (1979). 

That this condition is a generalization of the standard WKB (Wentzel-Kramers-Brillouin) condition can be seen by considering a one-degree-of-freedom system, where we have... 

The second condition is the generalized Brillouin theorem, first derived by Levy and Berthier. ... 

The XifS are the matrix dements of X with respect to the f s i, j being creation and annihilation operators for spinorbitals (pi and (pj. The condition for optimum orbitals is then the generalized Brillouin theorem ... 

It remains only to devise suitable methods of solving the MC SCF equations in one or other of the forms discussed above. We may distinguish three main families of methods (a) those that aim to satisfy the operator equations (8.2.8) (or the equivalent condition (8.2.31)), normally in a finite-basis form such as (8.2.12) (b) those that minimize the energy directly by using steepest descent or more general gradient techniques and (c) those that aim to satisfy the condition (8.2.38), which are usually described as Brillouin-condition methods. Many special techniques are available within each category the examples in the next three sections illustrate these main approaches. 

The Hartree-Fock state is thus characterized by a perfect balance between excitations and deexcitations for any pair of orbitals p and q, the interaction with the state generated by the excitation of a single electron from p to g is exactly matched by the interaction with the state generated by the opposite excitation. This result is known as the generalized Brillouin theorem (GBT) [1]. For closed-shell states, all interactions are trivially equal to zero (due to the structure of the Hartree-Fock state) except those with the singly excited states i a) and (10.2.19) then reduces to the special condition (10.2.17). For all other states, we may write the GBT (10.2.19) in the more explicit form... 

For a simple cubic metal the first zone is given by hkl = 100 (and equivalent values 010, 001, 100, 010 and 001). In the reciprocal lattice 100 is the point on the (reciprocal) x axis at a distance 1 a from the origin. The Brillouin zone is now produced as the locus of the general condition (Fig. 26a, drawn there for 110) ... 

Rather amusingly it turns out even at a very low level of description that there is a degree of concordance in general predictions concerning a class of conductive states at least for the class of "benzenoid" polymers. In particular within the framework of either the simplest Hiickel model or of the simplest resonance-theoretic rationale it seems that the same stmctural conditions arise for the occurrence of Peierls-distortion and the sometimes associated solitonic excitations. For the simple Hiickel model, starting with uniform P-parameters, such a structural condition is well-known [54-57] to be coimected with a 0-band gap for which the feimi energy Cp occurs at a rational multiple of the Brillouin-zone size, say at wave-vector k=Tip/q - then a distortion cutting the... 

We generally use a face-centred-cubic unit ceU Qattice constant 15.9 A] with constant volume [1000 A and periodic boundary conditions (PBC). This supert ll geometry leads to a weak interaction between the individual clusters, and the accurate reproduction of the symmetries in Sg, Sg and S12 [36], for example, shows that the results are insensitive to the choice of boundary conditions. The cut-off energy for the plane wave expansion (11) of the electronic eigenfunctions (10.6-14.0 Ry) leads to 4000-6000 plane waves for a single point (k = 0) in the Brillouin zone (BZ). 

This condition leads to a generalized form of the local Brillouin theorem of the usual SCF method... 

In other words, the stationary condition on V H ) is equivalent to a condition on the matrix elements connecting V with excited Vs formed by substituting an occupied spin-orbital by an unoccupied spin-orbital in all Slater determinants where it appears (destroying determinants in which it does not occur). This is reminiscent of the Brillouin theorem for a 1-determinant wavefunction, but is clearly a generalization it is not, however, in the form given by Levy and Berthier (1968). 

Equation (2.45) represents a speoial solution to the equation of motion (2.6). According to (2.33,35), there are N independent values of q lying in the first Brillouin zone -rr/a < q 4 ir/a. If q > 0, A(S) is the amplitude of a wave travelling to the right while A( S) is a wave travelling to the left. Since the two waves are independent, there are no necessary conditions between A(S) and A( S). The general solution will therefore be a superposition of solutions of the type (2.45), where the summation extends ever all modes... 

We start the discussion by formulating the Hamiltonian of the system and the equations of motion. The concept of force constants needs further examination before it can be applied in three dimensions. We shall discuss the restrictions on the atomic force constants which follow from infinitesimal translations of the whole crystal as well as from the translational symmetry of the crystal lattice. Next we introduce the dynamical matrix and the eigenvectors this will be a generalization of Sect.2.1.2. In Sect.3.3, we introduce the periodic boundary conditions and give examples of Brillouin zones for some important structures. In strict analogy to Sect.2.1.4, we then introduce normal coordinates which allow the transition to quantum mechanics. All the quantum mechanical results which have been discussed in Sect.2.2 also apply for the three-dimensional case and only a summary of the main results is therefore given. We then discuss the den-... 

The conditions described here also define the conditions for diffraction of electron waves at the Brillouin zone boimdaries. Likewise the Brillouin zones described in Chapter 2 are reciprocal-space objects with the symmetry of the reciprocal lattice rather than the real-space lattice. The reciprocal lattice points in Figure 2.5, for example, are located at points hbj, kb2, and lb3. The reciprocal lattice for a simple cubic system with basis vectors ai, a 2, and as has reciprocal lattice vectors parallel to the real space vectors. However, larger distances in real space correspond to shorter distances in reciprocal space. Thus, planes that are widely spaced in real space have closely spaced reciprocal lattice points and vice versa. One may determine by examination of Figure 4.2 that the (100) planes are V3 times farther apart than are the (111) planes. In general, the distance, d, between (hkl) lattice planes in a cubic system may be shown to be ... 

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